Self-affinity of non-convex quadrangles for all or sufficiently large dissection orders

Determine whether every non-convex quadrangle is self-affine; whether, for each non-convex quadrangle Q, there exists an integer n₀(Q) such that Q is n-self-affine for every integer n ≥ n₀(Q); whether this eventual self-affinity holds for every non-convex quadrangle; and whether there is a universal integer n₀ independent of Q for which every non-convex quadrangle is n-self-affine whenever n ≥ n₀.

Background

The paper establishes that non-convex quadrangles exhibit substantially different self-affinity behavior from convex quadrangles. Specifically, it constructs an n-self-affine non-convex quadrangle for every integer n ≥ 3 and proves that no non-convex quadrangle is 2-self-affine. It also shows that a constructed n-self-affine non-convex quadrangle is self-affine for the orders n+k(n−1), while leaving open the behavior of arbitrary non-convex quadrangles and of all sufficiently large dissection orders.

The unresolved questions ask whether self-affinity is universal among non-convex quadrangles, whether each individual non-convex quadrangle eventually admits self-affine dissections of every order, and whether such an eventual threshold can be chosen uniformly, analogous to the universal threshold n₀=5 known for convex quadrangles.

References

The previous results give a first access to the non-convex case, but leave many questions open. Is every non-convex quadrangle self-affine? Is there some non-convex quadrangle $Q$ and an integer $n_0=n_0(Q)$ such that $Q$ is $n$-self-affine for every integer $n \ge n_0$? Does the last apply to every non-convex $Q$? Does the last apply to every non-convex $Q$ with $n_0$ not depending on $Q$, as it does for convex $Q$ with $n_0=5$ \Theorem~2?

— Self-affine quadrangles  (2502.15521 - Richter et al., 21 Feb 2025) in Section Non-convex, final paragraph following the proof of Proposition (non-existence of 2-self-affine non-convex quadrangles)